E(2)-like Little Group for Massless Particles

For massless particles, Wigner observed the symmetry group is isomorphic
to the two-dimensional Euclidean group or E(2), consisting of one rotational
and two translational degrees of freedom. It is easy to associate the
rotational degree to the helicity. What about the translational degree of
freedom? This question was not completely addressed until 1990, 51 years
after 1939.

Thus, this is an interesting topic in history of physics. In 1939, Wigner
stated that the symmetry is isomorphic to E(2) and wrote down the matrix of
the form

This is the "ugliest" matrix in physics. Even these days, physicists
give up if the matrix is not unitary. I know this from my
own experience. Since 1973, my papers were based on non-unitary squeeze
matrices. Did you know that Lorentz boosts are squeeze transformations?
As soon as I start telling this, my colleagues run away from me.

It was Steven Weinberg who started tackling the above "ugly" matrix in
one of his 1964 papers. In 1979, I studied Weinberg's papers with
my younger colleagues, namely D. Han and D. Son, and came up with
this figure.

This two-dimensional figure tells about rotations and translations in the E(2)
plane, and the translations as gauge transformations. As for the issue
of gauge transformations, there are several other authors who said
the translations correspond to gauge transformations, and they were
quoted in my papers. Most certainly, I am not the first one to observe
this aspect of Wigner's E(2)-like symmetry.

After seeing this, I thought the person who would appreciate this most
was Eugene Wigner who wrote the above "ugly" matrix in his 1939 paper.
In 1985, I went to Princeton to tell him about the translation-like
variables in his E(2)-like group. He became very happy, but asked why
there have to be two gauge components ( x and y coordinates).

I then had to work hard to make Wigner happy. I pulled out Wigner's
1953 paper with Inonu on
group contractions, telling that a spherical
surface can become flat if the radius becomes large. Thus, we can
consider a tangential plane for the E(2) symmetry.

The E(2) symmetry comes from the plane tangential at the north pole,
while the cylindrical symmetry comes from the cylinder tangential at
the equatorial belt (left figure).
Click here for the 1987 paper by Kim and Wigner.
The four-by-four Lorentz transformation matrix produces a geometry which
deforms a sphere into an egg and a pancake, which eventually become
a cylinder and a plane respectively.
Click here for the 1990 paper by Kim and Wigner.

Indeed, both translation-like variables correspond to one up-down translation
on the cylindrical surface. Thus, Wigner's E(2)-like symmetry has one
rotational degree and one translational degree of freedom. This translational
degree of freedom corresponds to the gauge degree of freedom.

During the period of five years (1985-90), I went to Princeton frequently
to seek guidance from Eugene Wigner. I was like a graduate student working
on his thesis research under Wigner's guidance. When I was a "real graduate
student" (1958-61), I was afraid of him.

These days, I am frequently introduced as Wigner's youngest student, while my
thesis advisor was Sam Treiman when I
got my degree in 1961. I become angry when this introduction is based on my
photos with Wigner. I will be very happy if I am so introduced based on the
two papers quoted above.

Neutrino Polarization

These days, massless neutrinos are somewhat outdated. Yet, until
right-handed neutrinos are found, and until the speed of neutrinos are
found to be less than c, massless neutrinos will occupy an important place
in physics. Since 1956, they are known to be polarized. Before 1956,
neutrino polarizations were unthinkable because they violate the parity
invariance.

Now the question is whether neutrino polarization is a God-given natural
law or is derivable from a more fundamental physical principle. Instead
of resorting to God, let us ask Einstein. In 1905, Einstein formulated
special relativity with the Lorentz group as the fundamental space-time
symmetry of particles.

Among the transformations, in his classic paper of
1939, Eugene Wigner considered subgroups of the Lorentz group
which leaves the four-momentum of a given particle invariant. They are
called Wigner's little groups these days. These subgroups dictate the
internal space-time symmetries of particles in the Lorentz-covariant regime.

For a massive particle, Wigner's little group is locally isomorphic to
the three-dimensional rotation group. The particle can be brought to the
Lorentz frame in which it is at rest. In this frame, rotations leave its
four-momentum invariant.

For a massless particle, the little group is isomorphic to the
two-dimensional Euclidean group, consisting of rotations around the origin
and two translations. It is not difficult to associate the rotational
degree with the helicity of the massless particle. What happens to the
two translation degrees of freedom. They correspond to gauge degrees
of freedom. Wigner's original 1939 paper does not mention this.
After a stormy history, it was
finally determined in 1990 that these two translational-like
degrees of freedom collapse into one gauge degree of freedom. See also
the link on Wigner and Poincar&eacute.

For the spin-1 photon, its four-vector potential is gauge-dependent, and
is not directly observable. Its electromagnetic field is gauge-invariant
and is directly observable.

This figure is from Han and Kim, Phys. Rev. A vol. 37, 4494 (1988).

How can we talk about gauge transformations for massless spin-1/2 particles?
We can certainly define transformations of Wigner's little group. Consider
a neutrino whose momentum is along the z axis, as shown in the
figure. We can perform a rotation, boost, and another rotation to arrive
at the original rotation. The net effect is therefore a transformation of
Wigner's little group. The transformation matrix turns out to be a two-by-two
triangular matrix. The upper-right triangular matrix is the parity congugation
of the lower-left triangular matrix. The triangular matrix leaves one neutrino
spin invariant but changes the other spin. In this way, the neutrino is
polarized, and its anti-neutrino is polarized in the opposite direction.